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Arcwise-Connected

المؤلف:  Armstrong, M. A.

المصدر:  Basic Topology, rev. ed. New York: Springer-Verlag

الجزء والصفحة:  ...

14-7-2021

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Arcwise-Connected

Many authors (e.g., Mendelson 1963; Pervin 1964) use the term arcwise-connected as a synonym for pathwise-connected. Other authors (e.g., Armstrong 1983; Cullen 1968; and Kowalsky 1964) use the term to refer to a stronger type of connectedness, namely that an arc connecting two points  and  of a topological space  is not simply (like a path) a continuous function  such that  and , but must also have a continuous inverse function, i.e., that it is a homeomorphism between  and the image of .

The difference between the two notions can be clarified by a simple example. The set {a,b}" src="https://mathworld.wolfram.com/images/equations/Arcwise-Connected/Inline9.gif" style="height:15px; width:60px" /> with the trivial topology is pathwise-connected, but not arcwise-connected since the function  defined by  for all , and , is a path from  to , but there exists no homeomorphism from  to , since even injectivity is impossible.

Arcwise- and pathwise-connected are equivalent in Euclidean spaces and in all topological spaces having a sufficiently rich structure. In particular theorem states that every locally compact, connected, locally connected metrizable topological space is arcwise-connected (Cullen 1968, p. 327).

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REFERENCES:

Armstrong, M. A. Basic Topology, rev. ed. New York: Springer-Verlag, p. 112, 1997.

Cullen, H. F. Introduction to General Topology. Boston, MA: Heath, pp. 325-330, 1968.

Kowalsky, H. J. Topological Spaces. New York: Academic Press, p. 183, 1964.

Mendelson, B. Introduction to Topology. London, England: Blackie & Son, 1963.

Pervin, W. J. "Arcwise Connectivity." §4.5 in Foundations of General Topology. New York: Academic Press, pp. 67-68, 1964.